Aspects of Duality, Holography and Integrability

One of the greatest unsolved mysteries in physics is to understand Quantum Gravity - the description of the universe when the curvature of space and time becomes so extreme that classical General Relativity is no longer sufficient and quantum mechanics takes hold. String Theory has established itself as the foremost candidate for Quantum Gravity. Of its many successes perhaps the biggest is the idea of Holography. Holography says that certain Quantum Field Theories are equivalent, or dual, to a classical theory of Gravity living in a space time with one extra dimension. Within this framework very difficult calculations become much simpler classical geometrical problems in the dual holographic description. A remarkable feature is that in certain cases, Holography becomes "integrable". In general it requires advanced numerical tools to unravel the full dynamics of a physical system. However, in an integrable system, the high level of symmetries of the problem allow the equations describing it to be solved directly with analytic (or basic) mathematical techniques. A final aspect concerns the dramatic way, known as T-duality, that strings experience spacetime geometry. For strings, a large dimension in space time of radius R is equivalent to a small space of radius 1/R. This research project will bring together the themes of Holography, Integrability and T-duality to shed new light on the structure of String Theory and its relation to Quantum Field Theory.